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Show that the relation R defined in the set A of all triangles as R = {(T_1, T_2) : T_1 is similar to T_2}, is equivalence relation. Consider three right angle triangles T_1with sides 3, 4, 5, T_2 with sides 5, 12, 13 and T_3 with sides 6, 8, 10. Which triangles among T_1, T_2 and T_3 are related ?

Answer»

Solution :Here A = set of all triangles
and R =`{(T_1, T_2) : T_1` is similar to `T_2`}
`because ` Every triangle is similar to itself.
`therefore R` is reflexive.
Let `T_1, T_2 in A and (T_1, T_2 ) in R`
`therefore R` is symmetric.
Let `""T_1, T_2, T_3 in A`
and `""(T_1, T_2) in R and (T_2, T_3) in R`
`rArr T_1` is similar to `T_2 and T_2` is similar to `T_3`.
`rArr T_1` is similar to `T_3`.
`rArr (T_1, T_3) in R`
`therefore R` is tansitive.
`because R` is reflexive, symmetric and transitive.
`therefore ` R is an equivalence RELATION.
Now `"" (3)/(6) = (4)/(8) = (5)/(10) = (1)/(2)`
`rArr` The corresponding sides of `T_1and T_3` are proportional.
`therefore T_1 and T_3` are similar.
`rArr ` Triangle `T_1` is related to triangle `T_3`.


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